See Niddah 17a and Tosafos there ד"ה שורפן, which contrasts the tzaddik and the chassid as far as how they dispose of their fingernail cuttings (which can harm others). The tzaddik does what is necessary (in this case, burying the cuttings), while the chassid goes beyond that - burning them - to make absolutely certain that no such harm will occur, even though this is harmful to himself.

So it seems that a Tzaddik keeps all the appropriate rules, but a Chassid goes beyond the letter of the law to do extra good.

Interestingly, in Kallah Rabbati (5:2 or in some versions 8:9) the Beraita asks:

מאי צדיק ומאי חסיד? צדיק בעל צדקות, חסיד גומל חסדים. ‏

I'm not sure how to translate this exactly because it seems to be using the word in its definition. I could not find a commentary who explained the passage. Please leave suggestions in the comments below.

An example of this might be found in Shabbos 120a: אמר רבא חסידי אגרא דשבתא שקלי אלא אמר רבא הכא בירא שמים עסקינן ולא ניחא ליה דליתהני מאחרים ובחנם נמי לא ניחא ליה דליטרח והכי קאמר ואם היו פיקחין דידעי דכה"ג לאו שכר שבת הוא עושין עמו חשבון לאחר השבת. Rava says that if a person's property becomes ownerless as a result of a house fire on Shabbos, Chassidim will still save the owner's property and give it back to him for free after Shabbos. "Yirei Shamayim" will give it back too, but may demand a fee for their trouble.
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FredApr 30 '12 at 17:42